Let $R$ be a commutative ring and let $M$ be an $m\times n$ matrix with entries in $R$. $M$ defines a map $\widehat{M}:R[x_1,\ldots,x_n]\to R[y_1,\ldots,y_m]$ by extending the linear action on degree $1$ monomials multiplicatively. I'm interested in recovering the matrices representing the linear actions on higher degree monomials.
Specifically, fix a degree $d$ and consider the linear map $\widehat{M}_d:R[x_1,\ldots,x_n]_d\to R[y_1,\ldots,y_m]_d$ given by restricting $\widehat{M}$ to degree $d$ monomials. For some fixed monomial orders on the $x_i$ and $y_j$, $\widehat{M}_d$ can be represented by a $\binom{m+d-1}{d}\times\binom{n+d-1}{d}$ matrix with entries in $R$. I'd like to know how to find the matrix representing $\widehat{M}_d$ given $d$, $M$, and some monomial order (I'm non-prescriptive with the monomial order here because I'm not sure if there is a particular monomial order which makes the problem easier).
I can write a program that will solve this problem computationally, but I'm hoping there is a general solution.